III - VERSION
EXAM
1
1. For
A
= [~ ~]
and
B=[3 o -13-4 5]
find the entry in the second row and third column in the product AB.
b) -7
a) 11
2. Find
%
c)
-11
d) 7
in the following system
+ 2y % +y -3% + Y %
a)
-1
e) 3.
b) -2
c) 2
d) 3
2z
=5
5z
=6
=7
2z
e) none of the above.
3. Find the entry in the second row and first column of the inverse of the matrix
011
[12140 1]
a) 2
b) -2
c) 1/2
d) -1/2
37
e) none of the above.
Exam
III- Version 1
4. A set S is described by the inequalities given below. Find the y coordinate of the corner
point of S in the half plane x > o.
x+2~0
2x +y $ 8
3~- 5y$-1
a) 2
b) -2
c)
1
d)
S
has no corner points in the half plane x ~
O.
e) None of the above
5. A Leontief input-output model has two goods, steel and oil. The production processes
are such that .8 units of steel and .1 unit of oil are required to produce 1 unit of steel, and
.4 units of steel and .5 units of oil are required to produce 1 unit of oil. Suppose that the
economy is meeting an external demand of 2 units of steel and 1 unit of oil. Then
a) The level of production
b) The level of production
c) The level of production
d) The level of production
e) None of the above
of steel is at
of steel is at
of steel is at
of steel is at
two times the level of production of oil.
4.5 times the level of production of oil.
2.5 times the level of production of oil.
3.5 times the level of production of oil.
6. The Mythic Pizza Co. makes two types of pizza: the Milano and the Romano. Each
Milano pizza uses 9 oz. of dough and 14 oz. of cheese. Each Romano pizza uses 11 oz. of
dough and 6 oz. of cheese. On a certain business day they have 60lbs. (960 oz.) of dough
and 50 Ibs. (8000z.) of cheese available. They always wish to produce at least twice as
much Romano pizza as Milano pizza. The profit on each Milano pizza is $.40. The profit
on each Romano pizza is $.60.
SET UP the problem of finding the largest possible profit for this business day and
show your work
38
Exam III - Version
1
7. Find the entry in the first row and second column of the product A· A for the matrix
A=[~
a) 10
b) -18
c) 19
-;]
d) 31
e) 18 .
8. A lottery winner plans to invest part of her $90,000 in utility bonds and the rest in a
savings account. She wants to keep at least 1/3 of her money in utility bonds but wants
to have at least twice as much money in savings as in utility bonds. Set up equations or
inequalities to express her options. show your work; but do NOT solve
9. Find the entry in the second row and first column of the inverse of the matrix
-.3 -.2]
.4
[.4
a) 2
b) -2
c)
d) 3
.2
e)
-.3.
10. Suppose for tax purposes a $200,000 building is depreciated at a rate of 4 percent of its
original value per year and a $500,000 per year building is depreciated at a rate of 8 percent
of its original value per year. Let t represent the number of years that a building has been
depreciated and V be the value of the building after t years. Write linear equations relating
V and t for the two buildings. When do the two buildings have the same value? show
your work
r•••
-.,
,
, - --\
G :
M If-X
'I
• X I.
6)1
Sf.J~J
c-cr;-o
\ 13
'7~~
A I'lx.
~", +-61
If ~ i:>;
'Ix. +-11j' :!f9".D J
~ i'O()/
;(r7
V
==
9~ 0 o~ )( 4
-3
0,00 0;1
,~
~21JO 000 _ ;> 000 £;-1 v:::.
t: ::.trr I?
39
2.~ /
])
5'0000 -{;)- 'ICJlooo:e-)
EXAM
III - VERSION
2
1. Let
A=
0 2 2
(10422 3)
What is the entry in the second row and third column of A-I?
(a)
-!
(b)
!
(d) 1
(c) -2
(e) none of the others
2. Find all solutions of the system of equations with the augmented matrix
(b) x = 1,
(d) x = 1,
(a) x = 1, Y = 1, z = 0
(c) x = 1, y = 1 - z, z arbitrary
(e) none of the others
=
1,
y = 0,
y
z arbitrary
z = 0
3. For which value of k does the matrix A not have an inverse?
A=
(a) k
=1
(b) k
=2
(c) k
2
(13
=5
2 4
k
4
2 2)
(d) k
=6
(e) none of the others
4. Which of the statements given below accurately describes the set of solutions for the
system of equations with augmented matrix
(a)
(b)
(c)
(d)
(e)
the system has a unique solution
the system has no solution
the system has an infinite number of solutions and for all solutions x
the system has an infinite number of solutions and for all solutions z
none of the others
41
=2
=0
Exam
III - Version 2
5. Tom and Dick have a fruit juice stand at which they make both a sweet drink and
a tart drink. Each glass of sweet drink uses .8 of the juice of an orange and .2
of the juice of a lemon. Each glass of tart drink uses .6 of the juice of an orange
and .7 of the juice of a lemon. If they have 50 oranges and 40 lemons, how many
glasses of each type of drink should they make to use all of the oranges and lemons?
(a) 20 sweet, 40 tart
(b)25 sweet, 50 tart
(c) 15 sweet, 30 tart
(d) 30 sweet, 60 tart
(e) none of the others
6. Which of the following is not a corner point of the feasible set given by x ~ 0, y ~ 0,
2x + y ~ 4, x + 2y ~ 4, 2x + 2y ~ 5.
(a) (2,0)
(b) (0,2)
(e) none of the others
(c) (1,1)
7. Find the maximum of 2x - 3y on the feasible set given by x ~ 0, 2x
2x - y ~ 5.
(a) 0
(b) -5
(c) 15
+ 3y
8. Find the minimum of -x
x +2y > 4.
(a) -4
(b) -9
(d) no maximum exists
+
(e) none of the others
on the feasible set given by x ~ 0, y ~ 0, x
(c) no minimum exists
(d) 1
3y ~ 5,
+y
~ 3,
(e) none of the others
9. Let
A=
(-13 -4
2)
B =
1 -1
(40 -2
3) .
What is the entry in row 2 and column 3 of -2AB?
(a) -26
(b) -13
(c) -20
(d) AB is not defined
(e) none of the others
10. Which of the following matrices does not have an inverse?
(a)
2
(31 -4)
(e) none of the others
11. Which of the following lines has slope 2?
(a) x
=2
(b) y = 2
(c) x = 2y
+4
42
(d) 2x - y = 4
(e) none of the others
Exam
III- Version
2
12. Which of the following lines is parallel to the line through the points (1,3) and (2,6)?
(a) 3%- Y
=4
(b) %- 3y
=4
(c) y
=3
(d) %= 3
(e) none of the others
13. What is the value of y at the intersection of the lines 2%+ 3y
(b) ~
(a) (3,1)
+ 4y = 2?
in the feasible set given by the constraints
(c) (2,2)
(b) (3,-3)
-%
(e) none of the others
(d) ~~
14. Which of the following points is not
2%+ y ~ 4, %~ y, %+ y ~ 5.
= 6 and
(d) (2,0)
(e) none of the others
15. Let A be 3 x 4, B be 4 x 3, C be 3 x 3 and D be 4 x 4. Which of the following matrix
products is not defined?
(a) ABC
(b) BAD
(c) DBA
(d) CAB
(e) none of the others
16. Use Gaussian elimination to find all solutions of the following system of equations.
Show reduced form!
---
-4
-3
-5
4y
2
6y
+
w
z
4w
6w
5
% 2z
+1yw
+
2%
3%
%
8~
16:
B 1) C C-A-2J
X;-Sf21-Ifw
y:; ~IAMd
j) L
7JA-ZJ
'.f-3 w
tV :;: ~J-t fA ~
2-:;.
43
B~
EXAM III - VERSION 3
1. (4 points) Find the equation of the line through the point (4,7) that is parallel to the line through the
points (-5,10) and (4, -17).
2. (4 points) Sketch the graph of the set of solutions of the equation 4z
coordinate system below:
y
x
3. (4 points) Perform the following computation:
4 -7 -3 - 5 -2
[-69
42]
7 =
3-6 S]
[-S
4. (4 points) PerCorm the Collowingcomputation:
45
+ Sy + 6%= 24
on the Cartesian
Exam
III - Version
3
5. (5 points) Assume that A is a 2 x 3 matrix, B is a 3 x 2 matrix, C isa 3 x 3 matrix, and D is a 2 x 3
matrix.. In each of the following parts, circle the word Defined if all of the operations are defined, and
circle the word Undefined if any operation is undefined.
-
(a)
(b)
(c)
(d)
CA Undefined
ACD
AC
A2
3B+2C
Defined
(e)
6. (8 points) In each of the Collowing parts, circle Ye$ if the matrix is in reduced row echelon form, and
circle No if the matrix is not in reduced row echelon Corm.
021
3 01 0No
1 001
o10 01 Yes
o0000
~][~
(a)
(b)
(d)
(c)
[1
n
[0
20 0
46
Exam III - Version 3
7. (8 points) Let A. B. and X be the followingmatrices:
2 -1
-1 4]
7
A= I1
In each of &hefollowing parts. circle Yes if the column vector is a solution of the equation AX
and circle No if the column vector is not a solution of this equation.
(aJ
[lJ
y~
= B.
No
(d)
[
iJ
y"
No
In each of the Ilext four questions consider the system of linear equations that corresponds to the given
augmented matrix and decide which of the followingstatements is true.
(A)
(B)
(e)
(D)
(E)
(F)
(G)
The system
The system
The system
The system
The system
The system
None of the
bas no solution.
has exactly one solution.
bas exactly three solutions.
has an infinite set of solutions with exactly one arbitrary parameter.
has an infinite set of solutions with exactly two arbitrary parameters.
bas an infinite set of solutions with exactly three arbitrary parameters.
otbers is true.
8. (4 points)
o
o
Answer:
1
-1
9. (4 points)
Answer:
10. (4 points)
o
2
1 1
2
2
o
-2
Answer:
47
Exam III - Version 3
1
-1
-2
42points)
11. (4
Answer:
-2
~l ]
[ ~l
12. (9 points) The Nectar Nutrition Company produces three types of liquid dietary supplements: Normal,
High-Potency, and Lo-Cal. Producing 100 liters (l.) of Normal requires 16 liters of nectar A, 12 kilograms (kg.) of chemical B, 5 liters of preservative C, and 50 minutes for processing. Producing 100 liters
of High-Potency requires 12 liters of nectar A, 3 kilograms of chemical B, 4 liters of preservative C,
and 20 minutes for processing. Producing 100 liters of Lo-Cal requires 7 liters of nectar A, 9 kilograms
of chemical B, 6 liters of preservative C, and 40 minutes for processing.
(i) Formulate the process of determining the amounts of nectar A, chemical B, and preservative C, and
time for processing required to produce specified quantities of the three types of dietary supplements
as a matrix multiplication problem.
(ii) Use part (i) to determine the amounts of nectar A, chemical B, and preservative C, and time for
processing required to produce 600 liters of Normal, 400 liters of High-Potency, and 800 liters of
La-Cal.
13. (14 points) Use the Gauss-Jordan algorithm to find all solutions of the following system of linear equations:
-2z
3z
14. (14 points) Use the Gauss-Jordan
tions:
+ y-
=
- 2y+6z =
+ 4y - 5% =
Z
2%
9
-24
25
algorithm to find all solutions of the following system of linear equaZl
-2z1
+ 5Z2
- 10z2
- Z3 - 2Z4
+ 3Z3 + 5z4
48
= -11
= 27
Exam
III- Version 3
15. (10 points) The Always Dry Roofing Company produces three grades of asphalt roofing shingles: A, B,
and C (also known as 20-year, 2S-year, and 3D-year shingles, respectively). Producing 100 square feet
(sq. ft.) of type A shingles requires 1 pound of fine gravel, 2 pounds of fiber glass, and 4 pounds of coal
tar. Producing 100 square feet of type B shingles requires 2 pounds of fine gravel, 5 pounds of fiber
glass, and 8 pounds of coal tar. Producing 100 square feet of type C shingles requires 3 pounds of fine
gravel, 6 pounds of fiber glass, and 13 pounds of coal tar. The model for this process is
r2
[r1]
r3
=
2 5 6
%2,
[14 28 13
3] [:1:1]
:1:3
where
Z1
=
Z2
= amount
amount of type A shingles in units of 100 sq. ft.,
of type B shingles in units of 100 sq. ft.,
Z3 = amount of type C shingles in units of 100 sq. ft.,
r1 = amount of fine gravel in units of pounds,
r2 = amount of fiber glass in units of pounds, and
ra = amount of coal tar in units of pounds.
If the company has 330 pounds of fine gravel, 740 pounds of fiber glass, and 1,360 pounds of coal tar
available, how much of each type of shingle should the company produce to use all of these supplies?
(Use techniques discussed in lectures.)
49
10-13
Exam 3 - Version 1
This is a 75 minute test.
Part 1.
Problems 1 through 13 are worth 5 points each, a total of 65 points.
In each of the problems 1 through 4, the augmented matrix shown has been obtained by a sequence of row operations. In each case, decide which of the following statements is true about the
associated system of equations.
(A) The system has a unique solution.
(B) The system has no solution.
(C) The system has an infinite number of solutions with one arbitrary parameter.
(D) The system has an infinite number of solutions with two arbitrary parameters.
(E) None of the above.
4.3
-3
-4
0
-1
1ooo-11
2
-21 -81
0 -3
[I
[I
[I
0
0
0
6I
0
J]~]
5Ii]
j]
0
2
5. Find the maximum value of the objective function in the following linear programming problem.
Maximize 3x
+ y,
subject to
x2:0
y2:0
5x + 2y ~ 90
3x +4y ~ 96
<
,
(A) 96
(B) 51
(C) 54
(D) 30
20
y ::; 30
(E) none of the others
Exam 3 - \ersion 1
For problems 6, 7, and 8, the matrices A, B, and C are defined as
B=
A= [4
2 -5
-2 61]
3
[
1
C=
-12 -23]
-1
[
35
2
0
-3]
6. Find the (2, 1) element of AB.
(A) -2
(B) 8
(C) 12
(D) -5
(E) none of the others
7. Find the (1,2) element of2B - C.
(A) 7
8. A matrix
D satisfies
3B
(A) -1
(B) 5
(C) 9
+ D = 2C.
(B) 1
(D) 3
(E) none of the others
Find the (2,2) element of the matrix
(C) 2
(D) 3
D.
(E) none of the others
9. Which of the following systems of equations has a unique solution?
(A) { -4x
2x +
- 6y
3y
= 42
(B) { 4x
2x -- 6y
3y
(C) { 4x2x+-6y3y==
-42
(D) {
= 42
-4x2x+ -6y3y==-42
(E) none of the others
10. Find the y-coordinate of the intersection of the lines 3x - 4y = 11 and 5x + 2y
(A) -2
(B) 1 (C) 4 (D) 1/2 (E) none of the others
=
1.
11. below?
Which of the following constraints describes the shaded set shown on the coordinate system
(A) x - 2y :::;2
(D) x - 2y ~ -2
(B) x - 2y ~ 2
(E) none of the others
21
(C) x - 2y :::;- 2
Exam 3 - \ersion 1
12. A set S is described by the system of inequalities
2x-Y20
x -+ 3y
{ 6x
2y ::;
::; 15
10
Find the y-coordinate of the comer point of S in the first quadrant.
(A) 2
(B) 4
(C) 1
(D) 3
(E) none of the others
13. Find the equation of the line whose graph is shown below
=3
=-3
(A)x-y
(C)x-y
(B) 3x (D)x+y
y
=-3
=3
(E) none of the others
Part 2.
Problem 1 is worth 15 points, problems 2 and 3 are worth 10 points each, a total of 35 points.
1. Formulate
and solve the following problem. Show all work.
Barbara's Basket's, Inc., produces two types of decorative straw baskets, Colonial and Southwest
styles. One Colonial basket requires 30 pieces of yellow straw and 30 pieces of brown straw,
and one Southwest basket requires 50 pieces of yell ow straw and 10 pieces of brown straw Each
week the company has 4500 pieces of yellow straw and 1500 pieces of brown straw to be used
to produce baskets. There is a commitment to produce at least 30 Southwest baskets. The profit
is $4.00 for each Colonial basket and $6.00 for each Southwest basket.
How many baskets of each type should be produced each week to maximize profit?
2. Find all solutions of the folloWing system of equations. Show all work.
3y - z =-1
-x+2y2x- z5y=-1
=0
x -
{
22
Exam 3 - \ersion I
3. Find the inverse of the matrix A
=
0 1 1
. Show all work.
214
[101J
e
13 lJ /J-C
?avf2. :
1)
c 8 C/+-
W&!vI< ~
B I+--C
~
23
~Wh1.
Exam 3 - Version 2
This is a 50 minute test.
1. The line through (3,-1) and parall el to the line through the points (4,1) and (1,2) is given by
the equation
(A)x + 3y = 0 (B) -x + 3y = -6
(D) 3x
+y = 8
= C. SupposeB-l
2. Suppose A, B, and CarematncessothatAB
.
Then A =
(A)
29 17
-2
(D) 1154
(B)
6 30
= [21]
5 3
andC =
2 5
[03]
(C) [ -19
-15 86]
4
9]
-5]
(C)-3x + y = -10
(E) none of the others
[2 of
11]
(E) none
the others
3. Which of the following matrices are in reduced form?
B=
001
5 0
00001
[1 100
0]
(B) A only
(E) none of the others
(A) A and B only
(D) A, B, and C
4. The reduced form of the matrix
0
C=
1 1
3
0 1 5 0
000
1
o
0
0
0
11070]
(C) B only
is
2 155
[1134]
100
1
(A)
o 1 0 3
o 0 1 0
(D)
o
(B)
1 0 2 0
1 1 0
000
1
0 1 1 3
000
0
[1021]
(E) none of the others
5. An economy has two goods: steel and lumber. The production of 1 unit of steel requires .4 units
of steel and .9 units oflumber, and the production of 1 unit oflumber takes .2 units of steel and
.2 units of lumber. The external demand is 3 units of steel and 9 units of lumber. What is the
technology matrix A?
(A) [ ~]
(B) [:~
:n
(C)
U :;]
24
(D) [
;i]
(E) none of the others
.
Exam 3 - ~rsion 2
6.l~tA = [ ~
9 0 . The (1,1) entry of A -1 is
1
o2 0]
(A)-9 (B)1 (C)-1
(0) 9
(E) none of the others
7. Michigan Plant Co. sells two kinds of seed assortment packages: the regular mix and the decorative mix. The regular mix requires 4 ounces of vegetable seeds and 1 ounce of flower seeds.
The decorative mix requires 2 ounces of vegetable seeds and 3 ounces of flower seeds. In making the two mixes, they used up 5600 ounces of vegetable seeds and 340 ounces of flower seeds.
How many regular mix packages did they make?
(A) 150
8. Minimize lOx
+ 6y
(B) 80
(C) 16
(D) 100
(E) none of the others
in the region shaded below
6
5
4
Y3
2
1
(A) 10
(B) 94
(C) 120
(0) 20
(E) none of the others
9. Formulate as a linear programming problem, identifying variables, constraints, and the objective
function (DO NOT SOLVE): At Confused State University, there are' 55 professors and 210
AI.'s available to teach English 122. To teach a large section with 155 students, 1 professor
and 3 AI.'s are needed. To teach a small section with 28 students, 1 AI. is needed (and no
professor). Furthermore, at least 50% of the sections taught must be large. How many of each
type of section should be scheduled to maximize the number of students taught?
25
Exam 3 - \ersion 2
10. Find all solutions (if any) to the systems oflinear equations which correspond to the following
augmented matrices.
(a)
(b)
0 2
4
o 0
[505]
0
0 1 0
[1o 00 03
(c)
0 0 1
[ o1 01 00
3
02]'
4
95 ]
11. Graph and find the comer points of the set of points described by the inequalities
y~x+3
x~O
y~O ~ 5
{ x+y
Il-fr
2J
8
f-/1
CA-
0 1J
/)/D-1.(
~1
26
~
~w+t
EXAM III - VERSION 1
Find all solutions of the system of equations
x-2y=3
x+2y+2z=3
x+6y+4z=3
i)
ii)
Is it true that AA = -I? Justify your answer.
FindCA
,(',~"
·Find the inverse, if it exists, of the matrix
65
Exam III-Version
1
4. Suppose that a steel-coal model has a technology matrix A = [~
vector is [~]
~].
If the steel-coal demand
, find the production schedule which meets this demand.
5. Set up, but do not graph or solve, the following linear programming problem:
The California Dried Fruit Company prepares two types of dried fruit packages for sale during
the holiday season. The Special Pack contains 10 ounces of dates, 12 ounces of apricots, and 3
ounces of candied fruit. The Standard Pack contains 16 ounces of dates and 8 ounces of apricots.
The company has 1200 ounces of dates, 90 ounces of apricots, and 360 ounces of candied fruit.
The net profit is $5 for each Special Pack, and $2.50 for each Standard Pack. The goal is to
maximize profit.
6. Solve the following linear programming problem:
Find the x and y values which minimize
the quantity c = 5x + y subject to the constraints
x~O
y~O
8x+2y~90
3x+ 6y~60
66
EXAM III - VERSION 2
Multiple Choice (5 points each)
1
The line with equation 3x - 7y = -6 has x-intercept
a)
2
b) ~
c)
0
d) -2
e)
3
f) none of the others
2. The equation of the line passing through (-2,3) and parallel to the line passing through the
points (1,1) and (2,-2) is:
a) x+2y=4
d) 3x+y=-3
c) x-2y=-8
f) none of the others
b) 2x+y=-1
e) 4x+ y=-5
3. The system of two simultaneous equations x + 2y = 3 and 3x - 4y = 1 has solution vector X = [;]
given by:
a)
[~]
b)
[TI
d)
[i]
e)
[j]
4. For matrices A =
-1
1
f) none of the others
2
and B = -lIthe
[ -23 2]
2
column of the product matrix AB is:
entry in the second row and second
[ 21 -1
2 -1]
1
a) undefined
b) 5
d) 3
e)
c)
-8
2
f) none of the others
67
EXAM
III- VERSION
2
Multiple Choice (5 points each)
1
The line with equation 3x - 7y = -6 has x-intercept
a)
2
c)
b) ~
d) -2
0
e)
3
f) none of the others
2. The equation of the line passing through (-2,3) and parallel to the line passing through the
points (1,1) and (2,-2) is:
a
c) x-2y=-8
f) none of the others
b) 2x+y=-1
e) 4x+y=-5
a) x+2y=4
d) 3x+ y=-3
The system of two simultaneous equations x + 2y = 3 and 3x - 4y = 1 has solution vector X = [;]
given by:
a) [~]
d)
[1]
4. For matrices A =
-1
1
2
b)
[TI
e)
[j]
[ 21 -1
2 -1]
1
f) none of the others
and B = -lIthe
[ -23 2]
2
entry in the second row and second
column of the product matrix AB is:
a) undefined
b)
5
c) 2
d) 3
e)
-8
f) none of the others
67
Exam III-Version
2
5. Which set of inequalities describes the shaded region graphed below?
line 3x - 4y = 6
line ~
x+2y=4
(5,0)
x
a) y~ 0, x + 2y~ 4, 3x-4y~ 6
b) y~O, x+ 2y~4, 3x-4y= 6
c) x ~ 2, x + 2y ~ 4,3 x - 4y ~ 6
d) x ~ 2, x + 2y ~ 4, 3x - 4y ~ 6
e) x ~ 2, x + 2y~ 4, 3x- 4y ~ 6
f)
none of the others
6. Find the maximum of the function F(x,y)= 15x + 5y on the shaded region in problem 5;
namely, the shaded region is the feasible set for this function:
none
a)
Z3 of the others
35 2,of
c)1,c=-1
none
undefined
the others
ID
e) -13
-18
b)
1,c=2
a=
c=-1
00f)
a=
-1 C1= 3 0 5 1for2what
1 values of a and c is aA + cC =
8. If A = -1 1 2 2 and
a) a=2, c = 1
a) 17
[103]
68
5 1 12 0? 3 3
Exam III-Version
2
The Dynamite Pest Control Company manufactures regular and deluxe mousetraps. Each
regular mousetrap requires one ounce of metal and two ounces of wood, while each deluxe
mousetrap requires two ounces of metal and three ounces of wood in their manufacture. Each
day the company has available ten pounds of wood for manufacturing these mousetraps.
However due to an agreement with one of its suppliers at least five pounds of metal a day
should be used in making regular and deluxe mousetraps. Also the company must produce at
least twenty deluxe mousetraps daily.
9. Let x = the number of regular mousetraps made daily and y = the number of deluxe
mousetraps made daily. The constraint equations for this linear programming problem are
(lIb = 16 ozs):
S 80,
2x++3y
3yS~160
80
xd)+ x~20,y~0
2y
~
160,
2x
b)
f)
x~20,
y~O
x~30,y~0
a) x~0,y~20
10. The feasible set for the linear programming problem given in problem 9 has corner points:
a) (30,25), (80,0), (30,100/3)
b) (0,0), (30,40), (80,0)
c) (30,25), (0,80) , (30,100/3)
d) (0,40), (40,20), (50,20), (0,160/3)
e) (20,30), (80,0), (20,40)
f) none of the others
11 If the Dynamite Pest Control Corporation makes $2 profit on each regular mousetrap and
$3.50 profit on each deluxe mousetrap, the maximum daily profit for the linear
programming problem is:
a) $180
b) $5300
c) $160
d)
$3'}1)
e) $56013
f) none of the others
12. If a matrix has m rows and n columns, it is said to be of dimension mxn. If A has
dimension 5 x 9, B has dimension 3 x 7 and C has dimension 9 x 3, the dimension of ACB is:
a) 7x5
b) undefined
c) 5x7
d) 45 x 15
69
e) 63x20
f) none of the others
Exam III-Version
2
13. Consider the equations 2x - y + 3z = 1, -3y + x -z = 2 and 5x - z + 2y = 3. Writing these in
matrix form AX = B for the solution vector X = [~] the coefficient matrix A is:
-1
1 -1
5 5-1
2c)-3
b)
1 -2
e)
12-12-3
1-1
-1-1
~]
-13] 3]
o[2 [2
[ -1
~-13]
[ 2 -1 3]
14. A nursery sells baby garden bushes by charging a price which is a linear relation of the
height of the bush. If a bush 1.5 feet high sells for $38 and a bush 3 feet high sells for $68, for
how much will a bush 2 feet high sell?
b) $55
a) $48
e) none of the others
d) $42
c) $45
15. What is the equation of the line passing through the points (3, +9) and (0, O)?
a) 4x+y=O
b) 3x-y=0
c) x=3
d) y=-3
e) 3x+y=0
f)
none of the others
16. Which of the following system of equations has no solution?
a) 7x-2y=3
-14x+4y=~
b) 7x-2y=3
-14x+ 4y= 1
17. Which of the following system of equations has infinitely
a) 7x-2y=3
-14x+ 4y= 1
18. Let A =
[g~ ~]
b) 7x-2y=3
-14x +2y = 1
d) none of the others
c) 7x-2y= 3
-14x+2y=1
many solutions?
d) none of the others
c) 7x-2y=3
-14x+4y=~
be the technology matrix for a Leontief linear economic model. If the
production schedule vector X = [1~~]
a) [~]
b)
[~]
d) [~]
e)
[~]
, the demand vector D = [~]
f)
70
is:
none of the others
Exam III-Version
2
For the matrix A = [ 1
1 the second column for the inverse matrix A-I is (Hint: Calculate
11 001 -1
2]
the inverse matrix and save it for the next problem):
b)
d) [~]
m
e) [
f)
-tJ
none of the others
By using A-I which you calculated in problem 19, solve AX = B for the solution vector X
WbenB=[
a) [~]
n
b) [~]
e) [ -~
]
f)
71
none of the others
EXAM III - VERSION 3
~ (5 points) Find the equation of the line through the point (2,5) that is parallel to the line
through the points (-4,16) and (3, -26).
2. (5 points) Sketch the graph ofthe set of solutions of the equation 4x + y + 2z = 8 on the
cartesian coordinate system below:
y
OIl
••
x
~
3. (5 points) Perform the following computation:
••
••
-•
•
~
4.
(5 points) Perform the following computation:
}
'1
[5-2
3]
4 1 -6
i~
[2-3]
=
72
Exam /II-Version
3
5. (8 points) Assume that A is a 3 x 4 matrix, B is 4 x 2 matrix, C is a 2 x 4 matrix, D is a 4 x 2
matrix, E is a 2 x 2 matrix, F is a 4 x 1 matrix, and G is a 2 x 3 matrix. In each of the
following parts, circle the word "Defined" if all of the operations are defined, and circle the
word "Undefined" if any operation is undefined.
Undefined
Undefmed Defined
a) ABC
h) CB
G+
3B
D-5E
F
01
Yes
020 points)
03
421
No
No-1
-6. -1
(10
In eachYes
of the following parts, circle "Yes" if the matrix is in reduced row echelon
-1
01
-a
form, and circle "No" if the matrix is not in reduced row echelon form.
1J
~J
n
~J
G
73
Exam III-Version
3
7. (8 points) ~tA, B, and X be the following matrices:
A=
1
[ 21 -1 43 3]
B=
[i]
In each of the following parts, circle "Yes" if the column vector is a solution of the equation
AX = B, and circle "No" if the column vector is not a solution of this equation.
Yes
No
b) [
Yes
t]
No
Yes
-10
No
d) [ =i
3 ]
No
Yes
In each of the next four questions consider the system of linear equations that corresponds to the
given augmented matrix and decide which of the following statements is true.
a)
b)
c)
d)
e)
The system
The system
The system
The system
The system
£) The system
g) None of the
has no solution.
has exactly one solution.
has exactly four solutions.
has an infinite set of solutions with exactly one arbitrary parameter.
has an infinite set of solutions with exactly two arbitrary parameters.
has an infinite set of solutions with exactly three arbitrary parameters.
others is true.
8. (3 points)
Answer:
9. (3 points)
Answer:
74
Exam III-Version
3
10. (3 points)
Answer:
11 (3 points)
Answer:
12. (5 points) The inverse of the matrix A =
2 3 1
isA-l= -17 4 3
[112]
[28-511 -6-5]
3210
information to solve the following system of linear equations:
. Use this
x+y+2z
= 3
2x+3y+z
=-4
3x+~+ 10z = 2
13. (8 points) Find the inverse of the following matrix:
14. (8 points) The Yummy Baking Supplies Company produces three grades of yellow cake mix:
generic, normal, and deluxe. Each type of yellow cake mix requires flour, sugar, baking
soda, and yellow die number 5 (and some other ingredients also). Producing 100 pounds of
generic yellow cake mix requires 80 pounds of flour, 15 pounds of sugar, 2 pounds of baking
soda, and 6 ounces of yellow die number 5. Producing 100 pounds of normal yellow cake mix
requires 75 pounds of flour, 20 pounds of sugar, 3 pounds of baking soda, and 12 ounces of
yellow die number 5. Producing 100 pounds of deluxe yellow cake mix requires 70 pounds of
flour, 25 pounds of sugar, 3 pounds of baking soda, and 18 ounces of yellow die number 5.
Formulate the process of determining the quantities of flour, sugar, baking soda, and yellow
die number 5 required to produce specified amounts of the three kinds of yellow cake mix as
a matrix multiplication problem.
15. (10 points) Find all solutions of the following system of linear equations:
= 14
x+4y-3z
= -35
-2x-7y+11z
3x+ 19y + 27z = -9
16. (11 points) Find all solutions of the following system oflinear equations:
= -14
=-57
=00
75